This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space X. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology T on the measurable functions such that all the almost-everywhere convergence sequences converge in T, then all the convergent-in-measure sequences also converge in T.

Obvious questions are:

Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)

Interesting question, but I'm half tempted to mark it down for bad TeX. Please take a moment to learn about \langle and \rangle. (If it makes you feel better, many top mathematicians make the same mistake.)
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Harald Hanche-OlsenNov 14 '09 at 19:18

3

Since the question was just bumped anyway, I decided finally to fix the TeX issue Harald complained about.
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Mark MeckesOct 2 '12 at 13:15

A quick overview: On a set $X$, for every $x\in X$ we define which filters converge to $x$, with the following restrictions: the ultrafilter of all supersets of $x$ must converge to $x$; any filter which contains a filter converging to $x$ must converge to $x$; the intersection of two filters converging to $x$ must converge to $x$ («contains» and «intersection» to be understood in the usual set-theoretic sense).

Converging filters in a convergence subspace: a filter ${\mathcal F}$ converges to $x$ in the subspace iff the filter on the initial space generated by the filter basis ${\mathcal F}$ converges to $x$.

The sets which are present in every filter converging to $x$ are known as neighborhoods of $x$ with respect to the corresponding convergence structure $\Lambda$. Such sets are actually neighborhoods in the topological sense, for a topology on $X$ (called the topology associated with $\Lambda$). The definitions of a closed set as a set whose complementary is open, and as a set which coincides with its (filterwise) adherence, are equivalent. A set $K$ is said to be $\Lambda$-compact if every ultrafilter in $K$ converges with respect to the induced convergence structure on $K$. Filterwise continuous maps send compact sets onto compact sets.

A sequence in a topological space converges to a point iff for every open set around the point, cofinitely many elements of the sequence are in that open set. Let's fixate on the words "cofinitely many" in the previous sentence. The collection of cofinite sets is an ideal in a certain rig (ring without negation), namely the rig whose elements are subsets of the natural numbers, with multiplication given by union and addition given by intersection. One can replace this ideal with some other ideal and get a different notion of convergence. For instance, there is an ideal of all sets containing the number 4; using this ideal, sequence converge to their fourth elements. Alternately, you could extend the cofinite ideal, e.g. using axiom of choice find a maximal ideal containing it. If you use a maximal ideal, then every sequence in a compact space converges.

These turn out to be quite useful in studying smooth curves and differentiation in locally convex topological spaces. However, there won't be a topology with the property that either of these families is the family of convergent sequences.

For more on these, see A Convenient Setting of Global Analysis by Kriegl and Michor.

(I'm going to get a digital rubber stamp for the previous sentence to save me typing it every time.)

Maybe the most common form of non-topological convergence are Cesaro summability of series, and (for power series) Borel summability. The question was about sequences, not series. However, for sequences in a vector space, series and sequences are equivalent.

All of the examples in this thread are on the theme of sequences in a topological vector space. The set of such sequences is a new vector space, and the examples so far are, at the very least, shift-invariant linear extensions of the map $\lim$ from convergent sequences to their limits. Maybe any shift-invariant linear extension of $\lim$ could be called a theory of convergence? Let's say also that the extension should be continuous in some fairly fine topology on the sequences, maybe the box topology. If it is shift-invariant (with right shifts too if you pad with zeroes), then it automatically has the filter property that changing finitely many values does not change convergence.

Possibly linearity is not essential. You could look at shift-invariant extensions of convergence in a topological space which are again continuous in the box topology. Technically it should be a pointed space to define right shift, but the position of the point doesn't matter.

The following two books base Analysis on convergence structures (In German only).
It was realized early in the development of Calculus beyond Banach spaces that topology is not sufficient to express the usual approach to differentiability;
one ran into difficulties with the chain rule.
These books are an attempt to to remedy this.
(Many thanks to Andrew Stacey for his rubber stamp.)

Welcome to MathOverflow! That "digital rubber stamp" has been the source of many of my answers here (and, more importantly, a source of inspiration in my own work). I'll have to find another way to amass points here, but I'll happily make that trade for being able to read your answers to questions on this, and similar, topics. I look forward to learning from them.
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Loop SpaceOct 2 '12 at 11:52

Since that question popped up on the screen again I add something to the list (although I am pretty late to the party): The fact that pointwise almost everywhere convergence is not topological is nicely described in

There is a literature on "convergence spaces" of various kinds. I read some of that in the 70s, but I do not remember a lot of the detail. There is something called "pre-topological space" or "closure space". And there is "pseudo-topological space". Each of them can be defined in terms of convergence of filters. Or in terms of convergence of nets. Or in terms of neighborhood systems. Or in terms of a closure operation.. One of these is associated with Choquet. There is a big text Topological Spaces by Cech that takes the closure space as the fundamental notion.

Maybe I misunderstood, but if you mean Kuratowski definition of a topological space by means of closure, it is exactly equivalent to the usual definition by open, closed sets or neighborhood systems, and so cannot produce a larger class of "spaces".
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Gian Maria Dall'AraNov 15 '09 at 9:23

As one further example of this phenomenon, allow me to recall the (apparently forgotten) category of compactologies, which was introduced and studied by Buchwalter in the 60's. Roughly speaking, they are unions of compact spaces (with the appropriate compatibility conditions). Every topological space has a natural compactology and every compactological space has a natural topology, but the two categories are distinct. (They do coincide for nice spaces---metric spaces, locally compact spaces, ...). It is clear how to define convergence of sequences for compactological spaces. This category has two useful properties. Firstly, for many natural spaces, the canonical topology collapses them to a point but this doesn't happen with the compactological structure. Perhaps more importantly, this category has an intrinsic completeness concept (topological completeness is not intrinsic). Since, as a general rule of thumb, dual objects (at least, those of interest to analysts) are always complete, there will tend to be problems with duality theory if one uses topological spaces as a basis (we are thinking, in particular, of topics like the Gelfand-Naimark duality). In particular, we can never get a symmetric duality theory for all topological spaces (or rather completely regular spaces, since functional analysists, being interested in duality between underlying spaces and spaces of functions theoreon, tend to restrict attention to this case). In the context of compactological spaces, these problems are less acute.

I answer to the point (1). In all examples one addresses the question of

Convergence of a function (equivalently a family or a sequence) towards a certain point when the argument (equivalently the index of the family or the sequence) tends to some point.

In my experience, there are two ways of building a topologyless theory for the argument (my answer does not cover the value side, although all sets with a filter could be endowed with a topology: just consider the discrete topology on the set, add a point $\omega$ at infinity and consider the elements of the filter as its neighbourhoods, see e.g. Bourbaki General Topology 1).

Filters

Nets

Filters have been invoked in a previous answer, nets are just families $(x_i)_{i\in I}$ indexed with a (filtered or directed) ordered set $(I,<)$ (i.e. two elements have an upper bound). This point of view is strictly equivalent to that of the filters (easy, but too long, exercise), however, in practice the point of view of nets is more adapted to questions with algebraic manipulations whereas the one of filters is more convenient for questions where domains are involved (germs, jets, asymptotic scales etc.)